Many will be shocked when an assignment sheet asking you to “Measure the volume of an egg” is thrown at you out of the blue. Usually, when this question is asked, we will think of the following method:
- Measure and record the volume of water within a cup
- Immersing the egg into the water
- Measure and record the new volume of the water and egg
- then subtracting the new volume by the initial volume
Easy, but it definitely won’t get you an A.
So how else can we measure the volume of an egg?
Integration is a fundamental concept of calculus used as a method to find the formula of elements with the property of “instantaneous rate of change”. In shapes, such elements could be the integration of an equation to find the area beneath the curve of a function. In applications to real life, elements include the usage of the formula of acceleration to obtain velocity and displacement.
Amazingly enough, the applications of integration are not only limited to 2 dimensional objects. By using calculus, we are able to measure the volume of a real life egg – but how?
There are several methods and formulas that can be used to approximate the volume of an egg. The more precise your values are, the more credible the approximation will be.
The general concept of these methods will include:
- Draw the egg as a 2 dimensional image on a set of x and y axes (refer to diagram below)
- Calculate the area of half of the egg’s cross section (the area under the curve within the positive side of the y axis)
- Using integration to find the volume by using the concept of the rotation around the x axis (Integral of πy^2)
The first formula that can be used to calculate the area under the curve is the Simpson’s Rule:
You can also try to use excel to find the most accurate equation for your curve; usually the higher the power, the more accurate the equation will be.
A less accurate way to calculate the area is by using the Trapezoidal Rule:
The three different ways of calculating the area under the curve are all quite easy and simple.
So how about a challenge?
Teacher’s love originality. Reading and marking the same assignments with the same process and methods will seriously bore them out!
A more complicated and accurate way of calculating the volume of an egg is by thinking outside the box. If you look at the 2-dimensional egg drawn on the x and y axes, you can actually see it as a half of an ellipse (negative side of the x-axis) joined with a half of a circle (positive side of the x-axis). With this, you are able to use the general function of an ellipse and then integrate it to find the surface area underneath the curve of the ellipse! (Note: this is the area of the ellipse that is within the range of the positive y-axis).
The general function of an ellipse:
After the long and exciting integration, the area of the ellipse will be added the the area of the circle (using the well known πr^2), and can then be used to rotate around the x-axis, resulting in the volume of our dear egg.